Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions

Let C$C$ be a closed convex subset of a real Hilbert space H$H$ and assume that T$T$ is an asymptotically κ$\kappa$-strict pseudo-contraction on C$C$ with a fixed point, for some 0≤κ<1$0\le \kappa <1$. Given an initial guess _x0∈C$x_0\in C$ and given also a real sequence {_αn}$\left\{{\alpha }_n\right\}$ in (0, 1), the modified Mann’s algorithm generates a sequence {_xn}$\left\{x_n\right\}$ via the formula: _xn+1=_αnxn+(1−_αn)Tⁿ_xn$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T^n{x}_n$, n≥0$n\ge 0$. It is proved that if the control sequence {_αn}$\left\{{\alpha }_n\right\}$ is chosen so that κ+δ<_αn<1−δ$\kappa +\delta <{\alpha }_n<1-\delta$ for some δ∈(0,1)$\delta \in (0,1)$, then {_xn}$\left\{x_n\right\}$ converges weakly to a fixed point of T$T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.     

On Mann iteration in Hilbert spaces

Let K$K$ be a compact convex subset of a real Hilbert space H$H$; T:K→K$T:K\to K$ a hemicontractive map. Let {_αn}$\left\{{\alpha }_n\right\}$ be a real sequence in [0,1] satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the sequence {_xn}$\left\{x_n\right\}$ defined iteratively by _xn=_αnxn−1+(1−_αn)T_xn$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n$, n≥1$n\ge 1$ converges strongly to a fixed point of T$T$.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Factorization properties of paratopological groups

In this article we continue the study of R$\mathbb{R}$-factorizability in paratopological groups. It is shown that: (1) all concepts of R$\mathbb{R}$-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G   is R$\mathbb{R}$-factorizable if and only if it is totally ω  -narrow and has property ω-QU$\omega \text{-}\mathit{QU}$; (3) every subgroup of a _T1$T_1$ paratopological group G   is R$\mathbb{R}$-factorizable provided that the topological group G^?$G^{?}$ associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space  ; (4) if Π=_∏i∈IGi$\Pi ={\prod }_{i\in I}{G}_i$ is a product of _T1$T_1$ paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π   is R$\mathbb{R}$-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

On a Mann type implicit iteration process for continuous pseudo-contractive mappings

Let K$K$ be a nonempty closed convex subset of a Banach space E$E$, T:K→K$T:K\to K$ a continuous pseudo-contractive mapping. Suppose that {_αn}$\left\{{\alpha }_n\right\}$ is a real sequence in [0,1]$[0,1]$ satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the Mann type implicit iteration process {_xn}$\left\{x_n\right\}$ given by _xn=_αnxn−1+(1−_αn)T_xn,n≥0$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n,n\ge 0$, strongly and weakly converges to a fixed point of T$T$, respectively.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Nonparametric estimation for pure jump Lévy processes based on high frequency data

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n$n$ discrete time observations with step Δ$\Delta$. The asymptotic framework is: n$n$ tends to infinity, Δ=_Δn$\Delta ={\Delta }_n$ tends to zero while n_Δn$n{\Delta }_n$ tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global L²${\mathbb{L}}^2$-risk. Second, we use a direct approach (“time domain”) which allows us to construct an estimator on a given compact interval. We provide a bound for L²${\mathbb{L}}^2$-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators _Lα${\mathbf{L}}_{\alpha }$ in two variables, depending on the coupling parameter α?0$\alpha ?0$ that appears only in the boundary conditions. Our main concern is the spectral properties of _Lα${\mathbf{L}}_{\alpha }$, which turn out to be quite different for α<1$\alpha <1$ and for α>1$\alpha >1$. In particular, _Lα${\mathbf{L}}_{\alpha }$ has a unique self-adjoint realization for α<1$\alpha <1$ and many such realizations for α>1$\alpha >1$. In the more difficult case α>1$\alpha >1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.     

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

Nonparametric estimation for stochastic differential equations with random effects

We consider N$N$ independent stochastic processes (_Xj(t),t∈[0,T])$(X_j\left(t\right),t\in [0,T])$, j=1,…,N$j=1,\dots ,N$, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable _?j${?}_j$ and study the nonparametric estimation of the density of the random effect _?j${?}_j$ in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L²$L^2$-risk. Asymptotic properties are evaluated as N$N$ tends to infinity for fixed T$T$ or for T=T(N)$T=T\left(N\right)$ tending to infinity with N$N$. For T(N)=N²$T\left(N\right)=N^2$, adaptive estimators are built. Estimators are implemented on simulated data for several examples.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

On first-fit coloring of ladder-free posets

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w$w$ into w^14lgw$w^{14lgw}$ chains. They also observed that the problem of on-line chain partitioning of general posets of width w$w$ could be reduced to First-Fit chain partitioning of 2w²+1$2w^2+1$-ladder-free posets of width w$w$, where an m$m$-ladder is the transitive closure of the union of two incomparable chains _x1≤?≤_xm$x_1\le ?\le x_m$, _y1≤?≤_ym$y_1\le ?\le y_m$ and the set of comparabilities {_x1≤_y1,…,_xm≤_ym}$\{x_1\le y_1,\dots ,x_m\le y_m\}$. Here, we provide a subexponential upper bound (in terms of w$w$ with m$m$ fixed) for the performance of First-Fit chain partitioning on m$m$-ladder-free posets, as well as an exact quadratic bound when m=2$m=2$, and an upper bound linear in m$m$ when w=2$w=2$. Using the Bosek–Krawczyk observation, this yields an on-line chain partitioning algorithm with a somewhat improved performance bound. More importantly, the algorithm and the proof of its performance bound are much simpler.     

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).